Admissibility of local systems for some classes of line arrangements

arXiv:1207.4508v9 [math.GT] 15 Apr 2013

ADMISSIBILITY OF LOCAL SYSTEMS FOR SOME CLASSES OF LINE ARRANGEMENTS NGUYEN TAT THANG Dedicated to the memory of Dinh Thi Anh Thu. Abstract. Let A be a line arrangement in the complex projective plane P2 . Denote by M its complement and by M the set of points in A with multiplicity at least 3. A rank one local system L on M is admissible if roughly speaking the dimension of the cohomology groups H k (M, L) can be computed directly from the cohomology algebra H ∗ (M, C). In this work, we give a sufficient condition for the admissibility of all rank one local systems on M .

1. Introduction When M is a hyperplane arrangement complement in projective space Pn , the notion of an admissible local system L on M is defined in terms of some conditions on the residues of an associated logarithmic connection ∇(α) on a good compactification of M (see [9], [15], [10], [13] and [5]). This notion plays a key role in the theory, since for such an admissible local system L on M one has (1.1)

dim H k (M, L) = dim H k (H ∗ (M, C), α∧)

for all k ∈ N. Let A be a line arrangement in the complex projective plane P2 and denote its arrangement complement by M . For the case of line arrangements, a good compactification is obtained by blowing-up points of multiplicity larger than 2 in A. This explains the simple definition of the admissibility given below in Definition 2.1. In a recent paper [14], the authors introduced a class of line arrangements for which all rank one local systems on the complements are admissible. Namely, for each non-negative integer k, the line arrangement A is called to be of type Ck if k is the minimal number of lines in A containing all the points of multiplicity at least 3. It is proved in [14] that Theorem 1.1. Let A be a line arrangement in P2 . If A belongs to the class Ck for some k ≤ 2, then any rank one local system L on M is admissible. 2010 Mathematics Subject Classification. Primary 14F99, 32S22, 52C35; Secondary 05A18, 05C40, 14H50. Key words and phrases. admissible local system, line arrangement, characteristic variety, multinet, resonance variety. 1

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The purpose of this paper is to improve the work in [7], see Remark 2.6 below. More precisely, we give a combinatoric condition on a line arrangement A for the admissibility of rank one local systems on its complement M . Let M be the set of points in A with multiplicity larger than 2. Two points x, y ∈ M are called adjacent if they belong to a line H ∈ A (see [7]). Suppose that A satisfies the following condition: (C) For each point x ∈ M, there exist at most two lines H1 , H2 ∈ A such that x ∈ H1 ∩H2 and H1 , H2 contain all points in M which are adjacent to x. If all points of multiplicity ≥ 3 are situated on a line the arrangement is a nodal affine arrangement, see [3, 8]. Theorem 1.1 above shows that for such arrangements which are called of type C1 all rank one local systems on the complement are admissible. In [7] the author defines the notion of a path of length n to be the maximal sequence of distinct lines H1 , H2 , . . . , Hn where xi = Hi ∩ Hi+1 ∈ M and #{x ∈ Hi : x ∈ M} ≥ 2. One admits that lines in A containing only double points also make a path (i.e. the path {H}). If a point x ∈ M is not adjacent to any point in M and if H is a line passing through x, ′ ′ we consider {H} as a path and for all H passing through x, one identifies the path {H } with {H}. A path is called a cycle if H1 ∩ Hn ∈ M with n ≥ 3, otherwise it is called open (see [7]). Our first main result is the following. Theorem 1.2. Let A be a line arrangement in P2 satisfying condition (C). Assume that A has at most one cycle. Then all rank one local systems on the complement M of A are admissible. In particular, the characteristic variety V1 (M ) does not contain translated components and V1 (M ) is determined by the poset L(A). In Section 2 we first make explicit the admissibility condition in the case of line arrangements and recall the definition of characteristic varieties. Then we prove Theorem 1.2. In the end of Section 2 we give example of a line arrangement where the results in [7] and [14] can not be applied while Theorem 1.2 is useful (Example 2.8). In Section 3 we concentrate on arrangements having more than one cycle. The mains results in this section are Theorems 3.1 and 3.3 where we show that, under some additional assumptions, one still has the admissibility of all local systems. As an evidence, we give in Example 3.6 an arrangement and a nonadmissible local system on its complement. Accordingly, Theorem 1.2 does not hold if there are more than one cycle, also Theorem 3.3 is not true without the condition (1). That means our results are best possible. In the last section, we will study the multinets and resonance varieties. We will prove that if the line arrangements A satisfies the condition (C) then it does not support any multinets, equivalently, these is not any global resonance component except all lines in A are concurrent.

ADMISSIBILITY OF LOCAL SYSTEMS FOR SOME CLASSES OF LINE ARRANGEMENTS

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2. Admissible rank one local systems Let A = {H0 , H1 , ..., Hn } be a line arrangement in P2 and set M = P2 \ (H0 ∪ ... ∪ Hn ). Let T(M ) = Hom(π1 (M ), C∗ ) be the character variety of M . This is an algebraic torus T(M ) ≃ (C∗ )n . Consider the exponential mapping (2.1)

exp : H 1 (M, C) → H 1 (M, C∗ ) = T(M )

induced by the usual exponential function exp(2πi−) : C → C∗ . Clearly one has exp(H 1 (M, C)) = T(M ) and exp(H 1 (M, Z)) = {1}. More precisely, a rank one local system L ∈ T(M ) corresponds to the choice of some monodromy complex numbers λj ∈ C∗ for 0 ≤ j ≤ n such that λ0 ...λn = 1. A cohomology class α ∈ H 1 (M, C) is given by X dfj (2.2) α= aj fj j=0,n P where the residues aj ∈ C satisfy j=0,n aj = 0 and fj = 0 a linear equation for the line Hj . With this notation, one has exp(α) = L if and only if λj = exp(2πiaj ) for any j = 0, ..., n. Definition 2.1. A local system L ∈ T(M ) as above is admissible if there is a cohomology class α ∈ H 1 (M, C) such that exp(α) = L, aj ∈ / Z>0 for all j and for all point p ∈ H0 ∪ ... ∪ Hn of multiplicity at least 3 one has X aj ∈ / Z>0 . a(p) = j

Here the sum is over all j’s such that p ∈ Hj . For an admissible local system the isomorphism in (1.1) were proved in [9], [15]. Definition 2.2. The characteristic varieties of M are the jumping loci for the first cohomologies of M with coefficients in rank one local systems: Vik (M ) = {ρ ∈ T(M ) : dim H i (M, Lρ ) ≥ k}. When i = 1 we use the simple notation Vk (M ) = V1k (M ). Foundational results on the structure of the cohomology support loci for local systems on quasi-projective algebraic varieties were obtained by Beauville [2], Green and Lazarsfeld [12], Simpson [16] (for the proper case), and Arapura [1] (for the quasi-projective case and first characteristic varieties V1 (M )). Before prove Theorem 1.2 we introduce the following notion which is a generalization of a path as defined in the Introduction. Definition 2.3. A subset G := {Hi }i∈I ⊂ A is called a graph if it satisfies the following conditions: (i) For all i ∈ I then #{x ∈ Hi : x ∈ M} ≥ 2; (ii) For all i ∈ I, there exists j ∈ I \ {i} such that Hi ∩ Hj ∈ M;

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(iii) For any two points x, y ∈ M where x = Hi1 ∩ Hi2 , y = Hj1 ∩ Hj2 with some i1 , i2 , j1 , j2 ∈ I, there exists a path {Hk1 , . . . , Hkm }, ki ∈ I as defined in the Introduction such that x = Hk1 ∩ Hk2 , y = Hkm−1 ∩ Hkm . If H contains only one point x in M which is isolated (i.e. x is not adjacent to any point in M), we also call {H} a graph. If H does not contain any point of M we also admits that {H} is a graph. The graph G is said to be maximal if there does not exist H ∈ A \ G such that H ∪ G is a graph. For each graph G, we denote by {xi }i∈IG set of points in M which belong to all lines in the graph. One defines zone Z(G) associated to G as follows: Z(G) = {H ∈ A : ∃i ∈ IG , xi ∈ H}. Remark 2.4. (i) Every path is a graph. (ii) One deduces from the hypothesis (C) that the non-isolated point x ∈ M determines lines which contain x and points of M adjacent to x. Then, a graph G will be characterized by the set of intersection points xi ∈ M. Lemma 2.5. Let A be a line arrangement in P2 satisfying the condition (C). Then, the set of zones of all maximal graphs makes a partition of A. Proof. It is obvious that A = ∪G Z(G), where G runs over all maximal graphs of A. Now, let consider zones associated to two maximal graphs G1 and G2 . Assume that there exists H ∈ Z(G1 ) ∩ Z(G2 ). That means there are H1 ∈ G1 and H2 ∈ G2 such that x1 = H ∩ H1 ∈ M, x2 = H ∩ H2 ∈ M. If x1 6= x2 then H ∪ G1 and H ∪ G2 are graphs. Since Gi is maximal, one obtains that H ∈ G1 and H ∈ G2 . It implies that G1 ∪ G2 is also a graph. Hence G1 = G2 . If x1 ≡ x2 then H1 ∪ G2 is a graph. It deduces H1 ∈ G2 . Similarly, one obtains again that G1 = G2 . The proof is complete.  Proof of Theorem 1.2. Let L be a local system on M . In order to find a good cohomology class α for L, we will shape, graph by graph, the positive integer residues a(p) for all p ∈ M. Fix one line H0 in the cycle in A if it exists and any line in A containing at least two points in M, otherwise. We see from Definition 2.1 that the admissibility condition bases on the real part of the residues aH , H ∈ A. So instead of those complex residues, we may consider their real parts. It means we can assume that all residues aH , H ∈ A are real numbers.P Without loss of generality, we may assume aH ∈ [0, 1) for P all H ∈ A \ {H0 } and aH0 = − H6=H0 aH . Recall that for each x ∈ M we denote a(x) = H∈A,x∈H aH . Let G be any maximum graph. Case 1: H0 ∈ / G. We will correct aH , H ∈ G such that a(xi ) ∈ / Z>0 , i ∈ IG by several steps.

ADMISSIBILITY OF LOCAL SYSTEMS FOR SOME CLASSES OF LINE ARRANGEMENTS

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Step 1: Start with a line H1 ∈ G such that there is only one line H2 in G having intersection in M with H1 . Such a line exists since there does not exist any cycle in G. Let a1 := max{0, a(p) : p ∈ H1 ∩ M \ H2 , a(p) ∈ Z>0 }. Here and below, if the maximum is positive and attains at several points, we will take a1 as the sum a(p) at the point p which is not the intersection point of two lines in G. In this step, we replace aH1 by aH1 − a1 and aH0 by aH0 + a1 . It is obviously insures that X aH = 0 H∈A

and a(x) ∈ / Z>0 for all x ∈ H1 ∩ M \ H2 . Since a1 is either 0 or the sum of residues of some distinct lines H ∈ Z(G) with H 6= H0 one still has aH0 ≤ 0. Step 2: We continue with the line H2 defined in Step 1. Let a2 := max{0, a(p) : p ∈ H2 ∩ M \ (∪H∈G\{H1 ,H2 } H), a(p) ∈ Z>0 }. Denote by H2j , j ∈ J lines in G satisfying the following conditions: H2j 6= H1 , pj2 := H2j ∩ H2 ∈ M, a(pj2 ) ∈ Z>0 and a2 < a(pj2 ), ∀j ∈ J. We consider the following three possibilities. (a) #J ≥ 2 and aH = 0 for all H ∈ A \ G passing through some pj2 : One sees that a(pj2 ) = aH2 + aH j ∈ [0, 2). 2

a(pj2 )

It implies = 1 and hence a(p1 ) ∈ / Z>0 where p1 = H1 ∩ H2 . Then, we repeat the process from the beginning using the same method as in Step 1 for the maximal graph in G \ {H1 } which contains H2 (in this case, it is actually G \ {H1 }). ′ ′ (b) #J ≥ 2 and there exist H2 ∈ A \ G, j0 ∈ J such that pj20 ∈ H2 and aH ′ 6= 0 : Let 2

′

a2 := max{a(p) : p ∈ H2 ∩ M, a(p) ∈ Z>0 }. We replace aH2 by aH2 − a2 and aH ′ by aH ′ + a2 . Note that this does not change a(pj20 ) ′

′

2

2

but a(p) ∈ / Z>0 for all p ∈ H2 ∩ M \ H2j0 . Since aH ′ ∈ (0, 1) one still has aH ∈ / Z>0 for all 2 H ∈ A. In the next step, we continue with H2j , j ∈ J simultaneously. For each H2j we use the same method as we do with H2 . (c) #J ≤ 1: In this case, we correct residues as follows: aH2 := aH2 − a2 and aH0 := aH0 + a2 . It is easy to verify that a(p) ∈ / Z>0 for all p ∈ H2 ∩ M \ (∪i∈J H2j ). Similarly, in the next step we process with H2j , j ∈ J simultaneously. For each one, we repeat the method as in Step 2. We continue the process until the residue of all lines in the graph are corrected (may be changed or not). By the method we replace residues one can easily see that: Claim 1: a(p) ∈ / Z>0 , ∀p ∈ ∪H∈G H ∩ M.

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Claim 2:

P

H∈A aH

= 0.

In each step, we add to aH0 integer numbers which are either 0 or positive. In case of positive numbers, each of them has the form as follows: X a(p) = aH , p ∈ M. H∈A,p∈H

For H ∈ G we denote by bH the P origin residue of H (i.e. before replacements) and for x ∈ M denote by b(x) the sum x∈H bH . We shall prove the following. Claim 3: The sum A(G) we added to aH0 after correcting residue of all lines in G is X A(G) = bH , H

where H runs over some distinct lines in Z(G). Consequently, one has aH0 ≤ 0. Before proving Claim 3, we consider what we added to aH0 after first two steps: A2 := a1 + a2 . If a1 = 0 then A2 = a2 is either 0 or X

A2 =

bH ,

H∈Z(G),x∈H

for some x ∈ M. Otherwise, if a2 = a(p1 ), where p1 = H1 ∩ H2 then X a2 = (bH1 − a1 ) + bH . H∈Z(G),H6=H1 ,p1 ∈H

Therefore X

A2 =

bH .

H∈Z(G),p∈H

If a2 = a(q) for some q ∈ M \ H1 , it is easy to see the similar property of A2 . In order to show Claim 3 we write A(G) as follows: A(G) =

si m X X

ai,j

i=1 j=1

where ai,j ≥ 0, j = 1, . . . , si denote the integer numbers we added to aH0 in Step i when we correct residues of Hij (we rename lines whose residues were corrected in Step i by Hij ) and m is the number of steps we correct residues of all lines in G. One reminds that ai,j = max{0, a(x) : x ∈ Hij ∩ M \ (∪H∈G\(∪

j k k Hi−1 )∪Hi

It means ai,j is either 0 or ai,j =

X

H∈Z(G),x∈H

bH = b(y)

H), a(x) ∈ Z>0 }.

ADMISSIBILITY OF LOCAL SYSTEMS FOR SOME CLASSES OF LINE ARRANGEMENTS

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with some y ∈ Hij ∩ M \ (∪H∈G\{H j } H) or i

ai,j = (bH l

i−1

X

− ai−1,l ) +

bH

l H∈Z(G),H6=Hi−1 ,y∈H

l l where Hi−1 ∈ G has intersection in M with Hij and y = Hi−1 ∩ Hij ∈ M. In the last case, we see that ai,j + ai−1,l = b(y). Note that once we have ai,j in the last form, we also have the associated ai−1,l as a term of A(G). The corresponding between those terms is one-to-one due to the way of correcting residues. Now, we pair terms in A(G) as follows: We start with am,j , j = 1, . . . , m. If am,j has the last form, we pair it with the associated am−1,l , unless we keep it alone. By the same way, we continue with am−1,j which is not in a pair. We repeat the process until each of ai,j ’s is either in a pair or has one of the first two forms as above. Finally, one obtains that X A(G) = b(y),

where the sum takes over some distinct points y of ∪H∈G (H ∩ M). It is easy to check that there do not exist two points in y’s belonging to the same line in G. Claim 3 is proved. Our last claim is the following: P Claim 4: After replacing all residues we get B(G) := H∈Z(G) aH ≥ 0. P It is the consequence of Claim 3 and the fact B(G) = H∈Z(G) bH − A(G). Note that bH ≥ 0 for all H ∈ G. Case 2: H0 ∈ G. We correct residues aH of all lines in the graph G \ {H0 } (we may choose H0 such that G \ {H0 } is a graph) by the same way as in Case 1. By the same argument as above, we also receive properties as in the Claim (1-4). If x ∈ M is isolated then we replace aH by aH − a(x) and aH0 by aH0 + a(x), where H is any line containing x. To complete the proof, we need to check that a(x) ∈ / Z>0 for all x ∈ H0 ∩ M. Indeed, if x is not adjacent to any point out of H0 , according to Lemma 2.5, we have X X X XX aH ′′ ), aH + aH ′ + a(x) = − aH = −( H

I

x∈H /

where I runs over all graphs of A which do not contain H0 , for each I then H runs over ′ all lines in its zone; in the second term, H runs over all lines in the zone of the graph G \ {H0 }, where G is the graph containing H0 and the last sum takes over all lines which do not contain any point of M or contain only one point of M and have intersection in M with H0 . Each sum is non-negative according to Claim 4. Thus a(x) ∈ / Z>0 . ′ If x is adjacent to y ∈ H1 6= H0 . We have X X XX X a(x) = − aH = aH ′ − ( aH ′′ ). aH + aH ′ + 1

x∈H /

I

H

Because aH ′ < 1 and the sums are non-negative we obtain a(x) < 1. 1

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Finally, we obtain residues of all lines in A for L satisfying all conditions in Definition 2.1. In other words, the local system L is admissible. Combining this with results in [4] we get the properties of the characteristic varieties as shown in the theorem.  Remark 2.6. In [7], the author introduced a combinatoric condition for a line arrangement A such that all rank one local systems on its complement M are admissible. Theorem 2.7 (see [7]). Let A be a line arrangement in P2 satisfying condition (C). Assume that A has at most one cycle and on each line H ∈ A, there exist at most two points in M adjacent to points in M \ H. Then, all local systems on complement M of A are admissible. The following is an example of an arrangement in C3 for which both Theorem 1.1 and Theorem 2.7 cannot be applied, yet our new Theorem 1.2 shows that all local systems are admissible. Example 2.8. Let A be the arrangement in P2 defined by 13 lines: L0 : z = 0, L1 : x = 0, L2 : y = 0, L3 : x + 3y = 3z, L4 : 3y − x = 3z, L5 : x + 4y = 2z, L6 : x − 2y = 2z, L7 : x + y = 4z, L8 : 5y − 3x = 12z, L9 : 2x = z, L10 : y = 9z, L11 : y − x = 7z, L12 : y − x = 2z, see Figure 1, there are no parallel lines here. L9

L6

L11 p6

L0

p5

p4

p2

p3

p1 L4 L8

L5

L3 L7

L 12

L10

L2

L1

Figure 1. There are six points of multiplicity 3, these are p1 = [1 : 3 : 1] = L12 ∩ L7 ∩ L8 , p2 = [0 : 1 : 1] = L1 ∩ L4 ∩ L3 , p3 = [2 : 0 : 1] = L6 ∩ L2 ∩ L5 , p4 = [0 : 1 : 0] = L0 ∩ L1 ∩ L9 , p5 = [1 : 0 : 0] = L0 ∩ L2 ∩ L10 , p6 = [1 : 1 : 0] = L0 ∩ L11 ∩ L12 . Since there are 3 points p4 , p5 , p6 of L0 which are adjacent to other points in M \ L0 , the assumptions in Theorem 2.7 are not all satisfied. Also, since A is of type C3 then Theorem 1.1 can not be applied. However, one can easily check that the condition (C) defined in the

ADMISSIBILITY OF LOCAL SYSTEMS FOR SOME CLASSES OF LINE ARRANGEMENTS

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Introduction is fulfilled and there is not any cycle in A. Therefore, according to Theorem 1.2, all rank one local systems on the complement of A are admissible. 3. Admissibility for other classes of line arrangements In this section, we discuss the case where the arrangement has more than one cycle. One still has the admissibility of local systems provided some certain assumptions. Theorem 3.1. Let A be a line arrangement in P2 satisfying condition (C). Assume that all cycles in A have at least one line in common. Then, all rank one local systems on the complement M of A are admissible. Proof. We repeat the algorithm in the proof of Theorem 1.2 by firstly choosing H0 to be the common line of all cycles in A. The proof is then straightforward.  Let A be a line arrangement in P2 satisfying the condition (C). Denote by A1 set of all lines H ∈ A such that H contains only one point of M. Note that if M 6= ∅ then A1 6= ∅ (unless there exists x ∈ M and there are at least three lines passing through x which contain points adjacent to x, this contradicts to (C)). Let L ∈ T(M ) be a rank one local system and choose residues aH as in Definition 2.1 for H ∈ A. For each cycle C = {H1 , . . . , Hs } in A, we denote by PC the following subset of M: PC := {p1 , . . . , ps }, where pj = Hj ∩ Hj+1 for j = 1, . . . , s − 1 and ps = Hs ∩ H1 . Proposition 3.2. Let A be a line arrangement satisfying the condition (C) and fix a line H0 in A. Assume that for any cycle C in A not involving the line H0 there exists H ∈ A1 with H ∩ PC 6= ∅ such that aH ∈ / Z. Then, the local system L is admissible. Proof. By the same argument as in the beginning of the proof of Theorem 1.2, we may assume that aH ∈ [0, 1) for all H ∈ A \ {H0 }. Then the condition aH ∈ / Z means aH 6= 0. The idea of the proof is the same as in proof of Theorem 1.2, but firstly we open cycles in A. Let G be any graph of A. For each cycle C in G which does not contain H0 , according to the hypothesis, we can choose a line HC ∈ A1 such that aHC ∈ (0, 1) and HC passes through some point pC = HC1 ∩ HC2 ∈ PC , where HC1 , HC2 ∈ C. Let a := max{0, a(x) : x ∈ HC1 ∩ M, a(x) ∈ Z>0 }, where a(x) =

P

H∈A,x∈H

aH . Then, in the first step, we replace residues as follows: aH 1 := aH 1 − a, aHC := aHC + a. C

C

After this replacement, we see that aHC ∈ / Z>0 and a(x) ∈ / Z>0 for all x ∈ HC1 ∩ M \ {pC }. 1 However a(pC ) and aH with H ∈ / {HC , HC } do not change. Now we repeat the process as in the proof of Theorem 1.2 for each graph in ′

G := G \ (∪C {HC1 } ∪ H0 ). ′

During the process, we regard pC as a point of G of multiplicity at least 3 so that the corresponding residue a(pC ) is corrected when we shape the residue of HC2 .

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Finally, we obtain new residues with all conditions as in Definition 2.1 satisfied.



Theorem 3.3. Let A be a line arrangement satisfying the condition (C) and fix a line H0 in A. Assume that for any cycle C in A not involving the line H0 the followings hold: (1) The number of lines in C is even; (2) On each line H ∈ C, there exist at most two points in M adjacent to other points in M \ H. Then, all rank one local systems on the complement M of A is admissible. Proof. Let L be any rank one local system on M with residues aH , H ∈ A. Similarly, we may assume that aH ∈ [0, 1) for all H ∈ A, H 6= H0 . Let G be a maximal graph. If G contains H0 or G does not contain any cycle, we will shape residues of line in G \ {H0 } by using the same method as in the proof of Theorem 1.2. Since there is not any cycle in G \ {H0 } all Claims and argument there hold in this situation. Otherwise, according to the hypothesis, the graph G is itself a cycle which satisfies conditions (1) and (2) above, namely C = {H1 , . . . , H2k }. We consider the following possibilities. P (i) There exists a point p ∈ PC such that a(p) = H∈A,p∈H aH ∈ / Z>0 : Without loss generality, we can assume that p = H1 ∩ H2k . Then, we repeat the algorithm as in the proof of Theorem 1.2 for C by firstly correcting the residue of H1 : put a1 = max{0, a(x) : x ∈ H1 ∩ M \ H2 , a(x) ∈ Z>0 }. In the first step, replace aH1 by aH1 − a1 and aH0 by aH0 + a1 . We continue the process with H2 until residues of all lines are corrected. (ii) There exists H ∈ A1 , H ∩ PC 6= ∅ such that aH 6= 0: Using same method as in proof of Proposition 3.2. (iii) For all p ∈ PC then a(p) ∈ Z>0 and for all H ∈ A1 , H ∩ PC 6= ∅ we have aH = 0: In this case, due to aH ∈ [0, 1), H ∈ C then for p ∈ PC we obtain a(p) ∈ [0, 2), hence a(p) = 1. We replace residues as follows: aH2i := aH2i − a(p2i−1 ), aH0 := aH0 + a(p2i−1 ), i = 1, . . . , k, where p2i−1 = H2i−1 ∩ H2i ∈ PC . It is easy to see that after those replacements all Claims as in proof of Theorem 1.2 remain true. Thus we get new residues for L with all conditions as in Definition 2.1 satisfied. In other words L is admissible.  Let L be a rank one local system on the complement M of a line arrangement A and λH ∈ C∗ for H ∈ A be the corresponding monodromy numbers as in Definition 2.1. By the same argument as in the proof of Theorem 3.3 above, one can show the following. Corollary 3.4. Let A be a line arrangement satisfying the condition (C) and fix a line H0 in A. Assume that for any cycle C in A not involving the line H0 , on each line H ∈ C, there exist at most two points in M adjacent to other points in M \ H. Then, either L is admissible or there exists a cycle C such that λH = −1 for all H ∈ C and λH = 1 for all H ∈ / C which has intersection in M with some line of C.

ADMISSIBILITY OF LOCAL SYSTEMS FOR SOME CLASSES OF LINE ARRANGEMENTS

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Remark 3.5. In the following example, we will see that among arrangements satisfying the condition (C) one can not remove the assumption in Theorem 1.2 as well as the condition (1) in Theorem 3.3. Example 3.6. Let consider the arrangement A in P2 consists of 12 lines: L1 : x = 0, L2 : y = 0, L3 : x+ y − z = 0, L4 : x+ 3y = 0, L5 : x− 3y − z = 0, L6 : 3x− y + z = 0, L7 : x− y + 2z = 0, L8 : 4x+y−12z = 0, L9 : x+2y−10z = 0, L10 : x−y+8z = 0, L11 : 4x+y+12z = 0 and L0 : z = 0, this last one is the line at infinity. There are 6 points of multiplicity at least 3: p1 = [0 : 0 : 1] = L1 ∩ L2 ∩ L4 , p2 = [1 : 0 : 1] = L2 ∩ L3 ∩ L5 , p3 = [0 : 1 : 1] = L1 ∩ L3 ∩ L6 , p4 = [2 : 4 : 1] = L7 ∩ L8 ∩ L9 , p5 = [1 : 1 : 0] = L0 ∩ L7 ∩ L10 , p6 = [1 : −4 : 0] = L0 ∩ L8 ∩ L11 , see Figure 2, there are two disjoint cycles of length 3, without any line in common. We consider the rank one local system L = exp(α), where the cohomology class α ∈ 1 H (M, C) is given by residues ai := aLi = 1/2 for i ∈ {1, 2, 3, 7, 8}, aj := aLj = 0 for j ∈ {4, 5, 6, 9, 10, 11} and a0 := aL0 = −5/2. We will prove that this local system L is not admissible. L8 L7 L9

p4 L11

p5

L10 p6

p3 L4

L0 p1

L2

p2

L6

L5

L3

L1

Figure 2. Indeed, assume by contradiction that L is admissible. It means, there exists a coho′ mology class α ∈ H 1 (M, C) defined by residues bi := bLi ∈ C, i = 0, . . . , 11 such that P11 ′ exp(α ) = L, i=0 bi = 0, bi ∈ / Z>0 for any i and b(pj ) ∈ / Z>0 for any j = 1, . . . , 6, where X bk . b(pj ) = pj ∈Lk

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It is easy to see that bi = ki +1/2 for i ∈ {0, 1, 2, 3, 7, 8} and bj = kj for j ∈ {4, 5, 6, 9, 10, 11} with ki ∈ Z for all i ∈ {0, 1, . . . , 11}. Since bi ∈ / Z>0 we get kj ≤ 0 for j ∈ {4, 5, 6, 9, 10, 11}. We have the following equalities:   6 X X X bi  + b(pi ) = 2  bj i=1

i∈{0,1,2,3,7,8}

=−

X

j∈{4,5,6,9,10,11}

kj ≥ 0.

j∈{4,5,6,9,10,11}

P6

In other words i=1 b(pi ) ∈ Z≥0 . Moreover, one observes that b(pi ) is an integer for each i = 1, . . . , 6. Therefore b(pi ) = 0 for all i (since b(pi ) ∈ / Z>0 ) and hence kj = 0 for all j ∈ {4, 5, 6, 9, 10, 11}. In particular b1 + b2 = b2 + b3 = b1 + b3 = 0 so b1 = b2 = b3 = 0 which is impossible. Thus L is not admissible. 4. Multinets and resonance varieties In this section, we will work on the resonance varieties concerning our line arrangements and discuss how these resonance varieties behave. We use the notion of multinets which is defined in [11] where the authors gave the correspondence between the global components of the resonance varieties and the multinets. Definition 4.1. ([11]) A (k, d)-multinet on a line arrangement A is a partition A = ∪ki=1 (A)i of A into k ≥ 3 subsets, together with an assignment of multiplicities, m : A → Z≥0 , and a subset X ⊂ M of multiple points, called the base locus, such that: P (1) H∈Ai mH = d, independent of i; ′ (2) For each H ∈ Ai and H ′ ∈ Aj with P i 6= j, the point H ∩ H belongs to X; (3) For each X ∈ X, the sum nX := H∈Ai : H≤X mH is independent of i; (4) For each 1 ≤ i ≤ k and H, H ′ ∈ Ai , there is a sequence H = H0 , H1 , . . . , Hr = H ′ such that Hj−1 ∩ Hj 6∈ X for 1 ≤ j ≤ r. Lemma 4.2. Let A be a line arrangements in P2 such that the condition C is satisfied. Then, either all lines in A are concurrent or A does not support any multinet. Proof. Suppose that A supports a multinet A = ∪ki=1 Ai , k ≥ 3 with multiplicities m : A → Z≥0 and lines in A are not all concurrent. We denote by X the base locus. Let H1 ∈ A1 and H2 ∈ A2 arbitrary. According to the Condition (2) of Definition 4.1 ′ the point p := H1 ∩ H2 ∈ X. If p ∈ H for all H ∈ Ai , i > 2 there exists H ∈ A1 ∪ A2 ′ such that p ∈ / H . Unless, there is at least one line H ∈ Ai for some i > 2 where p ∈ / H. Anyway, there always exist at least 3 lines belonging to different sets of Ai ’s which are not concurrent. Without any loss, we call them by H1 ∈ A1 , H2 ∈ A2 , H3 ∈ A3 . According to the Condition(3) of Definition 4.1, number of lines in each Ai passing through each ′ point of X are the same. Therefore, there exist H3 ∈ A which passes through the point ′ q3 := H1 ∩ H2 ∈ X and H2 ∈ A2 passing through q2 := H1 ∩ H3 ∈ X. But it implies from ′ ′ the Condition (2) that H2 ∩ H3 ∈ X. Hence the Condition (C) fails. 

ADMISSIBILITY OF LOCAL SYSTEMS FOR SOME CLASSES OF LINE ARRANGEMENTS

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The (first) resonance varieties of A are the jumping loci for the first cohomology of the complex H ∗ (H ∗ (M, C), α∧), precisely: (4.1)

Rk (A) = {α ∈ H 1 (M, C) | dim H 1 (H ∗ (M, C), α∧) ≥ k}.

It is proved in [6] that the irreducible components of resonance varieties are linear subspaces in H 1 (M, C). A component R of R1 (A) is called a global component if R is not contained in any coordinate hyperplane (see [11]). Theorem 4.3. Let A be a line arrangements in P2 such that the condition C is satisfied. Then R1 (A) does not contain any global component except all lines in A are concurrent. Proof. This Theorem is a corollary of Lemma 4.2 and the following fact.



Theorem 4.4 ([11]). Suppose that the line arrangement A in P2 supports a global resonance component of dimension k − 1. Then A supports a (k, d)-multinet for some d. Acknowledgments The author would like to thank professor Alexandru Dimca for useful discussions and comments. The author would like to thank professor Sergey Yuzvinsky for the valuable comment about multinets and resonance varieties. References [1] D. Arapura: Geometry of cohomology support loci for local systems, I, J. Algebraic Geom. 6(1997), no. 3, 563-597. [2] A. Beauville: Annulation du H 1 pour les fibr´es en droites plats, in: Complex algebraic varieties (Bayreuth, 1990), 1-15, Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992. [3] A.D.R. Choudary, A. Dimca, S. Papadima: Some analogs of Zariski’s Theorem on nodal line arrangements, Algebraic and Geometric Topol. 5(2005), 691–711. [4] A. Dimca: On admissible rank one local systems, J. Algebra 321(2009), 3145-3157. [5] A. Dimca, L. Maxim: Multivariable Alexander invariants of hypersurface complements, Trans. Amer. Math. Soc. 359(2007), no. 7, 3505 - 3528. [6] A. Dimca, S. Papadima, A. Suciu: Topology and geometry of coho- mology jump loci, Duke Math. J., 148 (2009), no. 3, 405-457. [7] T. Dinh: Arrangements de droites et syst`emes locaux admissibles, unpublished, PhD Thesis, Nice, 2009. [8] T. Dinh: Characteristic varieties for a class of line arrangements, Canad. Math. Bull. 54(2011), no. 1, 56 - 67. [9] H. Esnault, V. Schechtman, E. Viehweg: Cohomology of local systems on the complement of hyperplanes, Invent. Math., 109 (1992), 557 - 561; Erratum, ibid. 112(1993), 447. [10] M. Falk: Arrangements and cohomology, Ann. Combin. 1(1997), no. 2, 135 - 157. [11] M. Falk and S. Yuzvinsky: Multinets, resonance varieties, and pencils of plane curves, Compositio Math. 143(2007), 1069-1088. [12] M. Green, R. Lazarsfeld: Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc., 4(1991), no. 1, 87- 103. [13] A. Libgober, S. Yuzvinsky: Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 121(2000), 337 - 361. [14] S. Nazir, Z. Raza : Admissible local systems for a class of line arrangements, Proc. Amer. Math. Soc. 137(2009), no. 4, 1307 - 1313.

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[15] V. Schechtman, H. Terao, A. Varchenko: Local systems over complements of hyperplanes and the Kac-Kazhdan condition for singular vectors, J. Pure Appl. Alg. 100(1995), no. 1-3, 93 - 102. ´ [16] C. Simpson: Subspaces of moduli spaces of rank one local systems, Ann. Sci. Ecole Norm. Sup. 26(1993), no. 3, 361- 401. Institute of Mathematics, 18 Hoang Quoc Viet road, Cau Giay District, 10307 Hanoi, Vietnam. E-mail address: [email protected]